Barrel chimneys typically served stoves rather than fireplaces. They were small, often with 8"x8" or smaller flues. To make them more substantial looking and safer they were thickened up with double wythe 8" thick walls where they passed through the roof. That gave them the appearance of having a substantial base at the roof line which a few courses higher narrowed to a shaft or "barrel" before flaring out again for the drip edge and decorative top.

One mason, Hamilton Wallace, who worked around Mosby Creek, Oregon in the late 1860's was also a preacher. He made his brick on site and specialized in barrel chimneys featuring various religious symbols (usually a cross) and had such skill that he usually drew a crowd, whereupon he would preach to those gathered from the rooftop.*
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*Phil Dole, Professor Emeritus, Univ. of Oregon

Larger chimneys serving fireplaces or chimneys with more than one flue rarely had a base or were increased in wall thickness as they passed through the roof. Probably they had enough mass to dissipate heat and so their mass made them safe, and they were already substantial so didn't need any augmentation for aesthetic reasons.